The document discusses the history and derivation of the Pythagorean theorem. It states that the theorem was first invented by the Greek mathematician Pythagoras, but was also discovered independently by ancient Indian and Babylonian mathematicians as early as 1900 BC. The document then provides the derivation of the theorem using similar right triangles, and proves the converse is also true. It concludes by discussing applications of the theorem such as determining if a triangle is right-angled or calculating unknown sides.

1. Pythagoras theorem its history and
its derivation
By
Mohammed Shoaib
Hansika Singh

2. INTRODUCTION
› Pythagoras theorem states that “In
a right-angled triangle, the
square of the hypotenuse side is
equal to the sum of squares of
the other two sides“. The sides
of this triangle have been named
Perpendicular, Base and
Hypotenuse.
› It was first invented by
Pythagoras of Samos.

3. Baudhayana sulabha sutra
› दीर्घचतुरस्रस्याक्ष्णया रज्ुुः पार्श्घमानी ततयघग् मानी
च
यत् पृथग् भूते क
ु रूतस्तदुभयं करोतत ॥
› This was the statement by
baudhayan one of the greatest
ancient Indian mathematician
› This statement stated that if
three squares are kept in the
form as shown is this figure
two perpendicular to each other
and the third one joining the
other twos then it is observed

4. PYTHAGORAS THEOREM INVENTION
› Around Two thousand five hundred years ago, a
Greek mathematician, Pythagoras, invented the
Pythagorean Theorem.
› Pythagoras theorem was invented in Babylon
and Egypt At about 1900 B.C.
› Some ancient clay tablets from babylonia
indicates that Babylonians in the second
millennium B.C, have invented the Pythagoras
theorem.

5. Pythagoras theorem derivation
› Proof:
› We know, △ADB ~ △ABC Therefore,
› AD/AB=AB/AC
› Or, AB2 = AD × AC ….……..(1)
› Also, △BDC ~△ABC
› Therefore,
› CD/BC=BC/AC
› (corresponding sides of similar triangles)
› Or, BC2= CD × AC ……………………………………..(2)
› Adding the equations (1) and (2) we get,
› AB2 + BC2 = AD × AC + CD × AC
› AB2 + BC2 = AC (AD + CD)
› Since, AD + CD = AC
› Therefore, AC2 = AB2 + BC2

6. Converse of Pythagoras theorem
› In a triangle if square of one side is equal to the sum of the squares of the other two
sides, then the angle opposite to first side is a right angle. n △EGF, by Pythagoras
Theorem:
› EF2 = EG2 + FG2 = b2 + a2 …………(1)
› In △ABC, by Pythagoras Theorem:
› AB2 = AC2 + BC2 = b2 + a2 …………(2)
› From equation (1) and (2), we have;
› EF2 = AB2
› EF = AB
› ⇒ △ ACB ≅ △EGF (By SSS postulate)
› ⇒ ∠G is right angle
› Thus, △EGF is a right triangle.
› Hence, we can say that the converse of Pythagorean theorem also holds.
› Hence Proved.

7. Application of Pythagoras theorem
It is used
• To know if the triangle is a right-angled triangle or not.
• In a right-angled triangle, we can calculate the length
of any side if the other two sides are given.
• To find the diagonal of a square.
Useful For
› Pythagoras theorem is useful to find the sides of a
right-angled triangle. If we know the two sides of a
right triangle, then we can find the third side.

8. Thank you